Mathematical Characterization of Universe Terminal Object: Unified Time Scale, Boundary Time Geometry and High-Dimensional Structure of QCA Universe

Ma hema ical Cha ac e iza ion o Uni e se Te minal
Objec : Unied Time Scale, Bounda y Time
Geome y and High-Dimensional S uc u e o QCA
Uni e se
Haobo Ma
1
Wenlin Zhang
2
1
Independen Resea che
2
Na ional Uni e si y o Singapo e
No embe 19, 2025
Abs ac
A he in e sec ion o gene al ela i i y, quan um eld heo y, and in o ma ion
heo y, inc easing wo k poin s o a common pic u e: all obse able s uc u es o he
physical uni e se can be iewed as "shadows" o some highe -dimensional ma h-
ema ical objec unde die en p ojec ions. Desc ip ions such as causal pa ial
o de s, space ime mani olds, sca e ing ma ices, bounda y algeb as, and quan um
cellula au oma a (QCA) a e me ely images o his high-dimensional objec in die -
en ca ego ies. In his pape , wi hin he amewo k o unied ime scale, bounda y
ime geome y, and QCA uni e se, we in oduce and cha ac e ize a "Uni e se Te mi-
nal Objec " o p o ide a p ecise deni ion o he "highes -dimensional ma hema ical
s uc u e o he uni e se".
Specically, we cons uc h ee ypes o uni e se ca ego ies wi h physical con-
s ain s: Ope a o Sca e ing Uni e se Ca ego y (
OpUni
), Bounda y Time Geom-
e y Uni e se Ca ego y (
GeoUni
), and QCA/Ma ix Uni e se Ca ego y (
QCAUni
).
These encode, espec i ely, he sca e ingspec al shi Wigne Smi h delay s uc-
u e unde unied ime scale, bounda y ime geome y wi h gene alized en opy and
quan um null ene gy condi ions, and disc e e QCA uni e se wi h ni e in o ma ion.
Fo each ca ego y, we cons uc a causal shadow unc o o he ca ego y o locally
ni e causal pa ial o de s (
Caus
), he eby o malizing he s a emen ha "causal
pa ial o de is me ely a shadow o high-dimensional s uc u e".
Based on his, we dene he "Causally Compa ible Uni e se T iple " ca ego y
(
Uni ⊂OpUni ×GeoUni ×QCAUni
), whose objec s a e iple uni e se objec s wi h
aligned causal shadows in
Caus
, and mo phisms a e s uc u e-p ese ing maps along
coa se-g aining di ec ions and co a ian on he causal side. Assuming he Unied
Time Scale Mo he Rule axiom, Fini e In o ma ion P inciple, and app op ia e
Chain Comple eness axiom, we p o e: he e exis s a unique (up o isomo phism)
e minal objec
Umax
in
Uni
, called he "Uni e se Te minal Objec ". The h ee
p ojec ions o his objec gi e he Ope a o Sca e ing Te minal Objec (
Omax
),
Geome icBounda y Te minal Objec (
Gmax
), and QCA Te minal Objec (
Qmax
),
which a e pai wise equi alen unde app op ia e b idging unc o s.
Fu he mo e, we p esen h ee s uc u al esul s. Fi s , any physically ealizable
uni e se model can be iewed as a unique p ojec ion o
Umax
in some ca ego ical
1
dual, hus causal pa ial o de s, small causal diamonds, and obse e wo ldlines a e
me ely shadows o
Umax
a die en p ojec ions and scales. Second, based on he
disc e e e sion o gene alized en opy and quan um ocusing conjec u e, we p o e
he "Small Diamond Renemen Theo em": unde ni e in o ma ion axioms, causal
small diamonds a any scale can be lled by amilies o smalle diamonds in a nea ly
en opy-addi i e manne ; he minimal physical s uc u e is de e mined only by he
in o ma ion cell scale, no by some xed geome ic minimal diamond. Thi d, ob-
se e s, memo y, and mul i-obse e consensus geome ies can be cha ac e ized in
Umax
as l e s on he causal shadow and consis en subobjec s in he h ee ep e-
sen a ions, ans o ming " ime delay equals memo y" and " olume is phenomenon
decoded om bounda y da a" in o p ecise heo ems o ela i e en opy and scale
densi y. Appendices p o ide de ailed p oo s o he exis ence and uniqueness o he
uni e se e minal objec , he small diamond enemen heo em, and he equi a-
lence o he h ee ep esen a ions.
Keywo ds:
Unied Time Scale; Bounda y Time Geome y; Quan um Cellula Au-
oma a; Causal Pa ial O de ; Small Causal Diamond; Te minal Objec ; Fini e In o -
ma ion P inciple; Holog aphic S uc u e
1 In oduc ion & His o ical Con ex
1.1 High-Dimensional S uc u e and "Shadow" Pe spec i e
Mode n physical heo ies o en swi ch be ween se e al non-equi alen " ep esen a ions":
1. Geome ic desc ip ion wi h space ime mani old
(M, g)
and causal s uc u e
J±
as
basic objec s;
2. Ope a o sca e ing desc ip ion wi h Hilbe space, ope a o algeb a, and sca e -
ing ma ix
S(ω)
as basic objec s;
3. Quan um Cellula Au oma a desc ip ion wi h disc e e la ice
Λ
, local Hilbe space
Hcell
, and local e olu ion
U
as basic objec s.
Meanwhile, he causal se p og am p oposes ha mic oscopic space ime is essen ially
composed o a locally ni e pa ial o de se , sugges ing ha e aining only "p ecedence
ela ions" can econs uc pa s o con inuous mani old s uc u e. The holog aphic p inci-
ple u he indica es ha deg ees o eedom in bulk egions can be encoded by bounda y
deg ees o eedom, wi h black hole he modynamics and he Bekens ein bound p o iding
specic en opy bounds o his "bounda y encoding".
These esul s collec i ely sugges : he so-called "Uni e se" can be unde s ood as
some high-dimensional s uc u e con aining all da a o causali y, geome y, ope a o s,
in o ma ion, and compu a ion, while he amilia space ime, elds, pa icles, and causal
pa ial o de s a e me ely p ojec ions o shadows o his high-dimensional s uc u e in
die en ca ego ies and scales.
This pape a emp s o axioma ize his pic u e: cons uc a ca ego y- heo e ic ame-
wo k simul aneously con olling unied ime scale, bounda y ime geome y, and QCA
uni e se, and dene and p o e he exis ence o a "Uni e se Te minal Objec " wi hin i ,
p ecisely equa ing he "highes -dimensional ma hema ical s uc u e o he uni e se" o
he e minal objec
Umax
o ca ego y
Uni
.
2
1.2 His o ical Backg ound: F om Causal Se s, Holog aphy o
QCA Uni e se
In space ime disc e iza ion a emp s, he causal se p og am p oposed by Bombelli
LeeMeye So kin assumes mic oscopic space ime essen ially consis s o a locally ni e
pa ial o de se , wi h con inuous mani old being me ely a coa se-g ained limi . This idea
p o ided a p eceden o "causali y- s " uni e se cha ac e iza ion. ([PhysRe Le ][1])
On he o he hand, he holog aphic p inciple and AdS/CFT duali y show ha g a i a-
ional deg ees o eedom in bulk a e equi alen o con o mal eld heo y on he bounda y,
na u ally encou aging ew i ing g a i a ional dynamics wi h bounda y algeb a, gene al-
ized en opy, and Quan um Focusing Conjec u e (QFC/QNEC). ([Re ModPhys][2])
Fu he mo e, Schumache We ne ga e s uc u al heo ems o e e sible Quan um
Cellula Au oma a, cha ac e izing hem as local uni a y ans o ma ions wi h ni e p op-
aga ion adius and ansla ion co a iance, and p o ing e e sible QCA ha e good classi-
ca ion and con inuum limi p ope ies. Ex ensi e wo k shows ha con inuum limi s o
a class o QCA can p oduce eec i e Di ac- ype eld heo ies and gauge eld heo ies,
p o iding igo ous ma hema ical suppo o "Uni e se as QCA". ([a Xi ][5])
On he sca e ing heo y side, EisenbudWigne Smi h delay ime and Bi man
K en o mula show ha sca e ing phase de i a i e, spec al shi unc ion, and ace o
Wigne Smi h g oup delay ma ix a e ela ed by a unied ela ion, in e p e ed as a single
ime scale densi y
κ(ω)
. This scale is connec ed o he gene a o o bounda y ime ans-
la ion in he bounda y Hamil onian o malism (B ownYo k quasi-local ene gy), he eby
uni ying sca e ing ime scale and bounda y ime geome y. ([A oms][11])
The abo e h eads indica e: he e should exis deep co espondences be ween sca e ing
ime scale, bounda y ime geome y, and QCA uni e se, whose common cons ain s can
be dis illed in o se e al axioms. This pape p oposes he concep o "Uni e se Te minal
Objec " on his basis.
1.3 Con ibu ions and Main Resul s
The goal o his pape can be summa ized as he ollowing ques ion: Unde unied
ime scale and ni e in o ma ion axioms, does he e exis a ma hema ical objec ha is
simul aneously maximal and consis en in ope a o sca e ing, bounda y geome y, and
QCA ep esen a ions, such ha all physically ealizable uni e se models can be iewed
as i s p ojec ions o coa se-g ainings? I i exis s, can his objec be cha ac e ized as a
e minal objec in some na u al ca ego y?
To answe his, his pape comple es he ollowing wo k:
1. Cons uc h ee ypes o uni e se ca ego ies wi h physical cons ain s
(OpUni ,GeoUni ,QCAUni )
,
and dene causal shadow unc o s o he ca ego y o locally ni e causal pa ial o de s
(Caus)
on each, o malizing he p oposi ion "causal pa ial o de is a shadow o high-
dimensional da a".
2. Dene "Causally Compa ible Uni e se T iple " ca ego y
(Uni)
, whose objec s a e
iple uni e se objec s wi h common causal shadow in h ee ep esen a ions, and mo -
phisms a e maps p ese ing s uc u e along coa se-g aining di ec ion and co a ian on
causal shadow.
3. Unde Unied Time Scale Mo he Rule , Fini e In o ma ion P inciple, and Chain
Comple eness axioms, use Zo n's Lemma o p o e he exis ence o a unique (up o isomo -
phism) e minal objec
(Umax)
in
(Uni)
, and p o ide i s h ee p ojec ions
(Omax, Gmax, Qmax)
3
and hei pai wise equi alence.
4. On he causal shadow
(Cmax)
o
(Umax)
, in oduce scale-pa ame e ized amilies o
small causal diamonds, and p o e "Small Diamond Renemen Theo em": any la ge-
scale small diamond can be lled by smalle -scale diamonds in a nea ly en opy-addi i e
manne ; he minimal physical uni is de e mined by in o ma ion cell scale a he han
geome ic diamond, igo ousizing he s a emen "small diamonds a e ne e minimal s uc-
u es, only sel -consis en s uc u es".
5. Cha ac e ize obse e s, memo y, and mul i-obse e consensus geome y as causal
l e s and subobjec s in
(Umax)
, p o ing quan i a i e co espondence be ween memo y
en opy along wo ldlines and unied ime scale, gi ing a p ecise heo em e sion o " ime
delay is equi alen o memo y".
The pape s uc u e ollows "Model & Axioms -> Main Theo ems -> P oo s -> Ap-
plica ions -> Enginee ing P oposals -> Discussion -> Conclusion -> Appendices".
2 Model & Assump ions
This sec ion p o ides unied ime scale axiom, ni e in o ma ion axiom, deni ions o
causal pa ial o de and small causal diamonds, and cons uc s h ee uni e se ca ego ies
and causal shadow unc o s, laying ounda ion o subsequen main heo ems.
2.1 Unied Time Scale Mo he Rule
Conside a class o sca e ing sys ems sa is ying s anda d aceable pe u ba ion condi-
ions, whose sca e ing ma ix can be w i en as equency-dependen uni a y ope a o
amily
S(ω)∈ U(H), ω ∈R.
Dene Wigne Smi h g oup delay ma ix
Q(ω) = −iS(ω)†∂ωS(ω).
Le
φ(ω)
be o al sca e ing phase,
ξ(ω)
be Bi manK en spec al shi unc ion,
ρ el(ω)
be co esponding ela i e densi y o s a es. Unde app op ia e egula i y and aceabili y
condi ions, s anda d ela ions hold:
φ′(ω) = π ξ′(ω), ξ′(ω) = ρ el(ω), ρ el(ω) = 1
2π Q(ω),
de i ed om Bi manK en o mula and spec al shi unc ion heo y. ([No es][10])
Acco dingly, in oduce unied scale densi y unc ion
κ(ω) = φ′(ω)
π=ρ el(ω) = 1
2π Q(ω).
Axiom A1 (Scale Iden i y)
Fo all ope a o sca e ing uni e se objec s conside ed in his pape , hei sca e ing
desc ip ions sa is y he abo e scale iden i y. In o he wo ds,
κ(ω)
is unique up o addi i e
cons an , se ing as he mo he densi y o global unied ime scale.
This axiom unies sca e ing phase de i a i e, spec al shi unc ion de i a i e, and
Wigne Smi h g oup delay ace in o a single ime scale densi y, p o iding a common
baseline o b idging sca e inggeome yQCA.
4
2.2 Fini e In o ma ion P inciple and Local Fini eness
Bekens ein bound and black hole he modynamics indica e ha o gi en ene gy and
spa ial scale, maximum en opy o a sys em has a ni e uppe bound. This inspi es he
ollowing in o ma ion- heo e ic axiom.
Axiom A2 (Fini e In o ma ion P inciple)
1. The e exis s a cons an
Imax ∈(0,+∞)
, such ha he loga i hmic ca dinali y o
any amily o physically dis inguishable uni e se s a es does no exceed
Imax
.
2. The e exis s a minimal eec i e in o ma ion cell scale
ϵ > 0
, such ha when
desc ip ion scale is smalle han
ϵ
, added de ails do no co espond o new physically
dis inguishable deg ees o eedom.
Fini e in o ma ion p inciple induces local ni eness on causal s uc u e.
Deni ion 2.1 (Locally Fini e Causal Pa ial O de )
A pa ial o de se
(X, ≺)
is called locally ni e i o any
x≺y∈X
, he closed
causal in e al
I(x, y) = {z∈X:x⪯z⪯y}
is a ni e se .
Local ni eness na u ally appea s in causal se s and disc e e QCA causal s uc u es.
Fini e in o ma ion p inciple ensu es any ni e space ime egion con ains only ni e " un-
damen al e en s", suppo ing physical a ionali y o local ni eness.
2.3 Causal Pa ial O de and Small Causal Diamond
Deni ion 2.2 (Causal Pa ial O de Ca ego y
Caus
)
Objec s o
Caus
a e all locally ni e causal pa ial o de s
(X, ≺)
, mo phisms a e o de -
p ese ing injec ions
: (X, ≺X)→(Y, ≺Y), x1≺Xx2⇒ (x1)≺Y (x2).
Deni ion 2.3 (Small Causal Diamond)
In
(X, ≺)∈Caus
, gi en
x≺y
, i closed in e al
I(x, y)
con ains no non- i ial
"b anching", i.e., he e exis s no
z=x, y
sa is ying
x≺z≺y
such ha
z
in oduces a
new causal b anch be ween
x, y
, hen
D(x, y) := I(x, y)
is called a small causal diamond.
On con inuous Lo en zian mani olds, small causal diamonds co espond o mic o
causal diamond egions cu by wo nea ly pa allel Null hype su aces, se ing as na u al
uni s o dening local g a i y and en opy condi ions. In locally ni e pa ial o de s,
he amily o small diamonds o ms a "decomposable uni " sys em, upon which he Small
Diamond Renemen Theo em will ely.
2.4 Ope a o Sca e ing Uni e se Ca ego y
OpUni
Deni ion 2.4 (Ope a o Sca e ing Uni e se Objec )
An ope a o sca e ing uni e se objec
O
is a quad uple
O= (H,A, S(ω); κ),
whe e:
1.
H
is a sepa able Hilbe space;
2.
A ⊂ B(H)
is a
C∗
-algeb a sa is ying locali y and quasi-locali y condi ions;
5

3.
S(ω)∈ U(H)
is a amily o sca e ing ma ices sa is ying app op ia e aceable
pe u ba ion and egula i y;
4.
κ(ω)
is scale densi y unc ion sa is ying Axiom A1.
Deni ion 2.5 (Ope a o Sca e ing Uni e se Mo phism)
Gi en wo objec s
Oi= (Hi,Ai, Si(ω); κi), i = 1,2,
a mo phism
:O1→O2
is a pai o maps
= (V, ϕ)
, whe e
V:H1→ H2
is isome ic
embedding,
ϕ:A2→ A1
is uni al
(∗)
-homomo phism, sa is ying:
1.
VA1V†⊂ A2
, compa ible wi h local s uc u e;
2.
S2(ω)V=V S1(ω)
almos e e ywhe e;
3.
κ1(ω) = κ2(ω)
almos e e ywhe e (allowing addi i e cons an ).
Objec s and mo phisms cons i u e ca ego y
OpUni
.
2.5 Bounda y Time Geome y Uni e se Ca ego y
GeoUni
Deni ion 2.6 (Bounda y Time Geome y Uni e se Objec )
A geome icbounda y uni e se objec
G
is gi en by da a
G= (M, g, ∂M;{A(B), ωB}B⊂∂M )
whe e:
1.
(M, g)
is a Lo en zian mani old sa is ying app op ia e ene gy condi ions and global
hype bolici y,
∂M
is i s bounda y;
2. Fo each bounda y pa ch
B⊂∂M
,
A(B)
is local obse able algeb a,
ωB
is co e-
sponding s a e;
3. Gene alized en opy
Sgen(B) = A ea(B)
4GN
+Sou (B)
and i s a ia ion along Null di ec ions sa is y Quan um Focusing Conjec u e (QFC) and
Quan um Null Ene gy Condi ion (QNEC).
4. Exis ence o bounda y Hamil onian o malism compa ible wi h scale densi y
κ(ω)
,
such ha B ownYo k quasi-local ene gy can be in e p e ed as conjuga e o bounda y
ime ansla ion.
Deni ion 2.7 (Geome icBounda y Uni e se Mo phism)
Mo phism
:G1→G2
be ween wo objec s
G1, G2
is a pai
= (Φ,{ψB}),
whe e
Φ : M1→M2
is causal-p ese ing local dieomo phism,
ψB:A2(Φ(B)) → A1(B)
is uni al
(∗)
-homomo phism, p ese ing mono onici y o gene alized en opy and quasi-
local s uc u e. Objec s and mo phisms cons i u e ca ego y
GeoUni
.
2.6 QCA/Ma ix Uni e se Ca ego y
QCAUni
Deni ion 2.8 (QCA Uni e se Objec )
A QCA uni e se objec
Q
is a quin uple
Q= (Λ,Hcell, U, |Ψ0⟩; Θ),
6
whe e:
1.
Λ
is coun able disc e e la ice se wi h ni e neighbo hood s uc u e;
2.
Hcell
is ni e-dimensional cell Hilbe space, o al space
HΛ=O
x∈Λ
Hcell;
3.
U:HΛ→ HΛ
is local uni a y e olu ion wi h ni e p opaga ion adius and ans-
la ion co a iance;
4.
|Ψ0⟩ ∈ HΛ
is ini ial cosmic s a e;
5.
Θ
is uni e se pa ame e ec o , encoding local coupling and opological da a,
compa ible wi h scale densi y
κ(ω)
in app op ia e con inuous limi .
Deni ion 2.9 (QCA Uni e se Mo phism)
Gi en
Qi= (Λi,Hcell,i, Ui,|Ψ0,i⟩; Θi), i = 1,2,
a mo phism
:Q1→Q2
is gi en by iple
= (ι, χ, W)
whe e
ι: Λ2→Λ1
is la ice embedding o coa se-g aining map,
χ:Hcell,2→ Hcell,1
is cell
Hilbe space embedding,
W:HΛ1→ HΛ2
is isome ic embedding o con ac ion map
compa ible wi h
Ui
. Objec s and mo phisms cons i u e ca ego y
QCAUni
.
2.7 Causal Shadow Func o s
In oduce unc o s o
Caus
on h ee uni e se ca ego ies, o malizing "causal pa ial o de
is me ely pa p ojec ion o high-dimensional s uc u e".
Deni ion 2.10 (Causal Shadow Func o s)
1. Fo
OpUni
, dene
Fop :OpUni →Caus, O 7→ (XO,≺O),
whe e
XO
is se o esol able sca e ing e en s a chosen obse a ion esolu ion,
≺O
induced by posi i e/nega i e s uc u e o g oup delay ma ix
Q(ω)
and ligh cone condi-
ions.
2. Fo
GeoUni
, dene
Fgeo :GeoUni →Caus, G 7→ (XG,≺G),
whe e
XG
is ep esen a i e se o space ime e en s,
≺G
induced by
J±
causal eachabili y;
ni e in o ma ion p inciple ensu es local ni eness.
3. Fo
QCAUni
, dene
Fqca :QCAUni →Caus, Q 7→ (XQ,≺Q),
whe e
XQ= Λ×Z
a e disc e e space ime e en s,
≺Q
gi en by disc e e ligh cone s uc u e
wi h ni e p opaga ion adius.
One can e i y di ec ly ha hese h ee shadows cons i u e co a ian unc o s, p ese -
ing coa se-g aining mo phism di ec ion. Causal pa ial o de hus becomes he common
"shadow laye " o h ee uni e se desc ip ions.
7
3 Main Resul s (Theo ems and Alignmen s)
This sec ion in oduces causally compa ible uni e se iple ca ego y
Uni
based on abo e
models and axioms, and p esen s main heo ems on exis ence and uniqueness o uni e se
e minal objec , as well as s uc u al conclusions on small diamond enemen , obse e s,
and mul i- ep esen a ion alignmen .
3.1 Causally Compa ible Uni e se T iple Ca ego y
Uni
Deni ion 3.1 (Uni e se T iple )
A uni e se iple objec is a iple
U= (O, G, Q),
whe e
O∈Obj(OpUni ), G ∈Obj(GeoUni ), Q ∈Obj(QCAUni ),
sa is ying causal compa ibili y: he e exis s a locally ni e causal pa ial o de
C= (X, ≺
)
, and isomo phisms in
Caus
αop :Fop(O)∼
−→ C, αgeo :Fgeo(G)∼
−→ C, αqca :Fqca(Q)∼
−→ C.
Call
C
he causal shadow o
U
, deno ed
Fcaus(U) = C
.
Deni ion 3.2 (Uni e se T iple Mo phism)
Gi en
Ui= (Oi, Gi, Qi)
(
i= 1,2
), a mo phism
:U1→U2
is a iple
= ( op, geo, qca),
whe e
op :O1→O2, geo :G1→G2, qca :Q1→Q2
a e mo phisms in espec i e ca ego ies, sa is ying: on causal shadow side, h ee shadow
mo phisms
Fop( op), Fgeo( geo), Fqca( qca)
a e isomo phic in
Caus
o he same o de -p ese ing map
Fcaus(U1)→Fcaus(U2)
.
Deni ion 3.3
Objec s and mo phisms as abo e cons i u e ca ego y
Uni
, called Causally Compa ible
Uni e se T iple Ca ego y.
3.2 Renemen P eo de and Maximal Consis en Objec
In oduce " enemen " p eo de on
Uni
o compa e "in o ma ion ichness" o die en
iple s.
Deni ion 3.4 (Renemen Rela ion)
Fo
U1, U2∈Obj(Uni)
, i he e exis s mo phism
:U1→U2
, say
U1
enes
U2
,
deno ed
U1⪯U2
. I
U1⪯U2
and
U2⪯U1
, say
U1, U2
a e isomo phic, deno ed
U1≃U2
.
Modulo isomo phism,
⪯
educes o pa ial o de .
Deni ion 3.5 (Maximal Consis en Uni e se T iple )
8
I
U∈Obj(Uni)
sa ises: once
U⪯V
hen necessa ily
U≃V
, hen call
U
a maximal
consis en uni e se iple .
In ui i ely, a maximal consis en objec allows no addi ion o new compa ible s uc-
u es in h ee ep esen a ions. Uni e se e minal objec will be a ca ego y- heo e ic
s eng hening o maximal consis en objec .
3.3 Chain Comple eness Axiom
To use Zo n's Lemma, need comple eness axiom.
Axiom A3 (Chain Comple eness)
Fo any o ally o de ed sub amily (chain)
C ⊂ Obj(Uni)
in
Uni
, he e exis s a uni e se
iple
UC
such ha o any
U∈ C
,
U⪯UC
, and is a leas uppe bound in his sense.
Physically,
C
can be unde s ood as " enemen sequence" adding mo e sca e ing da a,
geome ic de ails, o QCA ules. Axiom A3 equi es hese enemen s o piece oge he
in o a consis en uni e se iple in he limi . This equi emen can be gua an eed by
aking weak limi s o app op ia e opological limi s o ope a o s, me ics, and QCA
ules in componen ca ego ies, and using con inui y o causal shadow unc o s and ni e
in o ma ion p inciple.
3.4 Exis ence and Uniqueness o Uni e se Te minal Objec
Wi h abo e p epa a ions, we s a e he main heo em.
Theo em 3.6 (Exis ence and Uniqueness o Uni e se Te minal Objec )
Unde Axioms A1A3 and Fini e In o ma ion P inciple A2, he e exis s a maximal
consis en objec
Umax
in ca ego y
Uni
, sa is ying:
1. Fo any
U∈Obj(Uni)
, he e exis s a unique mo phism
U:U→Umax;
2. I ano he objec
U′
max
also sa ises abo e p ope ies, hen
U′
max
is isomo phic o
Umax
.
Thus
Umax
is a e minal objec o
Uni
, unique up o isomo phism.
This heo em ans o ms he in ui i e s a emen "highes -dimensional ma hema ical
s uc u e o uni e se exis s and is unique" in o a igo ous ca ego y- heo e ic p oposi ion.
De ailed p oo in Appendix A.
3.5 Te minal Objec Images and Alignmen in Th ee Rep esen-
a ions
Le na u al p ojec ion unc o s be
Πop :Uni →OpUni ,Πgeo :Uni →GeoUni ,Πqca :Uni →QCAUni ,
dene
Omax = Πop(Umax), Gmax = Πgeo(Umax), Qmax = Πqca(Umax).
Theo em 3.7 (Maximal Consis ency and Alignmen in Th ee Rep esen a-
ions)
1.
Omax
is maximal consis en objec in
OpUni
;
(Gmax, Qmax)
a e maximal consis en
objec s in
(GeoUni ,QCAUni )
espec i ely;
9
cons an , e c., bu specic alues and es able p edic ions equi e de i a ion in specic
sub-models.
8 Conclusion
This pape in oduces "Uni e se Te minal Objec "
Umax
wi hin unied ime scale, bound-
a y ime geome y, and QCA uni e se amewo k, and p o es i s exis ence and uniqueness
in causally compa ible uni e se iple ca ego y
Uni
unde na u al axioms. I s h ee p o-
jec ions
(Omax, Gmax, Qmax)
a e maximal consis en in espec i e uni e se ca ego ies and
pai wise equi alen unde b idging unc o s, p o iding clea ca ego ical cha ac e iza ion
o "highes -dimensional ma hema ical s uc u e o uni e se".
On causal shadow
Cmax
o
Umax
, Small Diamond Renemen Theo em shows: causal
diamonds a any scale can be lled by smalle diamonds in nea ly en opy-addi i e man-
ne ; minimal physical uni de e mined by in o ma ion cell scale, no geome ic diamond.
Causal pa ial o de s, small diamonds, and obse e wo ldlines a e hus shadows o
Umax
a specic p ojec ions/scales; sca e ing delay, memo y en opy, and gene alized en opy
mono onici y a e mani es a ions o same mo he scale
κ(ω)
in die en ep esen a ions.
This amewo k bases u u e wo k: sys ema ic s udy o black hole en opy/in o ma ion,
cosmological cons an / acuum ene gy, quan um chaos/ETH, s ong CP/axion, g a i a-
ional wa e dispe sion/Lo en z iola ion wi hin
Umax
, seeking docking wi h specic ob-
se a ions and expe imen al pla o ms.
Acknowledgemen s
Au ho s hank esea ch in sca e ing spec al heo y, quan um in o ma ion, gene al el-
a i i y, and quan um eld heo y o p o iding solid ounda ion o in eg a ing unied
ime scale, bounda y ime geome y, and QCA uni e se.
Code A ailabili y
Pape is mainly axioma ic and heo e ic. Simula ion codes o QCA con inuous limi s,
disc e e causal se cons uc ion, and nume ical e alua ion o gene alized en opy will be
made a ailable a e unied o ganiza ion.
Re e ences
[1] L. Bombelli, J. Lee, D. Meye , R. D. So kin, "Space- ime as a causal se ", Phys.
Re . Le . 59, 521 (1987).
[2] R. Bousso, "The holog aphic p inciple", Re . Mod. Phys. 74, 825 (2002).
[3] J. D. Bekens ein, "Black holes and en opy", Phys. Re . D 7, 2333 (1973).
[4] J. D. B own, J. W. Yo k, "Quasilocal ene gy and conse ed cha ges de i ed om
he g a i a ional ac ion", Phys. Re . D 47, 1407 (1993).
16

[5] B. Schumache , R. F. We ne , "Re e sible quan um cellula au oma a", quan -
ph/0405174 (2004).
[6] T. Fa elly, "A e iew o quan um cellula au oma a", Quan um 4, 368 (2020).
[7] R. Bousso, Z. Fishe , S. Leichenaue , A. C. Wall, "A Quan um Focussing Conjec-
u e", Phys. Re . D 93, 064044 (2016).
[8] R. Bousso, E. Tabo , "Disc e e Max-Focusing", JHEP 06 (2025) 240.
[9] K. B. Sinha, "Spec al shi unc ion and ace o mula", P oc. Indian Acad. Sci.
(Ma h. Sci.) 104, 819853 (1994).
[10] F. Gesz esy, "Applica ions o Spec al Shi Func ions", lec u e no es (2017).
[11] P. C. Deshmukh e al., "EisenbudWigne Smi h ime delay in a omlase in e ac-
ion", and R. Shaik e al., "EWS Time Delay in Low Ene gy e-C
60
Elas ic Sca e -
ing", A oms 12, 18 (2024).
[12] F. Hiai, "Concise lec u es on selec ed opics o on Neumann algeb as",
a Xi :2004.02383.
A Appendix A: De ailed P oo o Exis ence and Unique-
ness o Uni e se Te minal Objec
This appendix gi es ull p oo o Theo em 3.6.
A.1 A.1 Pose Cons uc ion and P epa a ion o Zo n's Lemma
Le
O
be objec se o
Uni
, dene equi alence ela ion
U1∼U2
i exis s isomo phism
U1→U2
. Le quo ien se
P=O/∼
, deno e equi alence class
[U]
.
Dene pa ial o de on
P
[U1]≤[U2]
i
∃ :U1→U2.
This deni ion is independen o ep esen a i e choice. T ansi i i y and eexi i y hold.
An isymme y gua an eed by "di ec ed isomo phism implies isomo phism": i
U1→U2
and
U2→U1
, hey induce bidi ec ional enemen in h ee ep esen a ions, implying
equi alence in each, and equi alence o causal shadows, nally
U1≃U2
. Thus
(P,≤)
is
pose .
A.2 A.2 Exis ence o Chain Uppe Bound (Implemen a ion o
Axiom A3)
Le
C ⊂ P
be a chain, choose ep esen a i e amily
{Ui= (Oi, Gi, Qi)}i∈I
. Cons uc
limi objec in each ep esen a ion ca ego y:
1. **Ope a o Sca e ing**: Unde app op ia e opology (e.g. weak ope a o opol-
ogy), ake closu e o inc easing union o
Ai
as
A∞
, ake weak limi o
Si(ω)
as
S∞(ω)
.
Con inui y o spec al shi and scale densi y ensu es limi sa ises scale iden i y.
17
2. **Geome icBounda y**: Use G omo Hausdo  and weak-
∗
con e gence o
me ics
gi
and bounda y algeb as
Ai(B)
, ci ing s abili y esul s o gene alized en opy
and QFC/QNEC in limi s, ob aining limi objec .
3. **QCA**: Use local quasi-local opology o cellula ules and ini ial s a es, en-
su ing consis ency wi h con inuous limi and unied scale b idge.
4. Causal shadow unc o s p ese e co a iance and con inui y, so shadows align in
limi o locally ni e pa ial o de
C∞
.
Thus ob ain
U∞∈Obj(Uni)
, and mo phisms
Ui→U∞
, making
[U∞]
uppe bound o
C
.
A.3 A.3 Applica ion o Zo n's Lemma
By A.2, e e y chain in
(P,≤)
has uppe bound. Zo n's Lemma implies exis ence o
maximal elemen
[Umax]∈ P
. Pick ep esen a i e
Umax
, maximal consis en uni e se
iple .
A.4 A.4 Maximal Consis ency implies Te minal Objec P ope y
**Exis ence**: Le
U∈Obj(Uni)
. I no mo phism
U→Umax
, conside se
S={V∈Obj(Uni) : ∃ :V→Umax}.
Cons uc new objec
W
con aining
U
and
Umax
ia amalgama ion: e.g., on ope a o side
ake minimal algeb a con aining bo h on consis en causal shadow. This ensu es
U⪯W
and
Umax ⪯W
, and
[W]>[Umax]
, con adic ing maximali y. Thus mo phism exis s.
**Uniqueness**: I wo mo phisms
1, 2:U→Umax
exis , cons uc equalize
˜
U
.
˜
U
enes
U
and
Umax
, bu is s ic ly smalle han
Umax
, con adic ing maximali y. Thus
1= 2
.
A.5 A.5 Uniqueness o Te minal Objec
I ano he e minal objec
U′
max
exis s, unique mo phisms
:Umax →U′
max
and
g:
U′
max →Umax
exis . Composi ions mus be iden i y mo phisms, so
Umax ≃U′
max
.
B Appendix B: Equi alence o Te minal Objec Images
in Th ee Rep esen a ions
P oo o Theo em 3.7.
B.1 B.1 Componen Maximal Consis ency
Take
Omax
. I
O∈OpUni
exis s wi h
Omax ⪯O, O ≃ Omax
, cons uc
U′= (O, Gmax, Qmax)
.
Omax ⪯O
implies mo phism
Umax →U′
. Maximali y implies
[Umax] = [U′]
, so
Omax ≃O
,
con adic ion.
18
B.2 B.2 Sca e ingGeome y B idging
Cons uc
Φop→geo :O7→ G
using unied ime scale and bounda y Hamil onian o malism
o econs uc geome y/en opy om sca e ing. Cons uc
Φgeo→op :G7→ O
using
bounda y algeb a/modula ow o econs uc sca e ing/
S(ω)
. On physical subca ego y,
Φgeo→op ◦Φop→geo ≃Id
, e c.
B.3 B.3 Geome yQCA B idging
QCA con inuous limi app oxima es eld heo y/geome y. Use his o cons uc
Ψqca→geo
.
In e se
Ψgeo→qca
cons uc s disc e e QCA app oxima ion on geome y. Equi alence holds
on app oximable subca ego y.
C Appendix C: De ailed P oo o Small Diamond Re-
nemen Theo em
P oo o Theo em 3.8.
C.1 C.1 Scale Calib a ion
In
Gmax
, choose local coo dina es app oxima ing Minkowski. Fo scale
, dene small
diamond
Dp, =J+(p−)∩J−(p+)
. P ojec o
Cmax
ia
Fgeo
. Fini e in o ma ion p inciple
A2 gi es minimal scale
min
.
C.2 C.2 Fini e Co e ing
Fo
2> min
and
Dp, 2
, choose in e nal poin s
{pi}
o co e wi h
Dpi, 1
(
1< 2
). Local
ni eness ensu es ni e co e ing suces.
C.3 C.3 App oxima e Addi i i y o Gene alized En opy
1. A ea e m: sum o a eas app oxima es o al a ea, e o om o e laps con olled by
cu a u e and hickness
O(ε( 1, 2))
.
2. En opy e m: s ong subaddi i i y gi es
Sou (Dp, 2)≤PSou (Dpi, 1) + Eo
, e o
anishes as
1/ 2→0
. QFC ensu es mono onici y along Null di ec ions, bounding e e se
inequali y.
C.4 C.4 Limi
min
ensu es en opy bound unde xed ene gy/ olume. Limi
1→ min
alid.
19
D Appendix D: Fu he Cla ica ion on Obse e Fil-
e s and Memo y En opy
D.1 D.1 Obse e Fil e s
Obse e l e
FO
on
Cmax = (X, ≺)
sa ises l e axioms (uppe closed, in e sec ion
closed). Gua an ees obse e can s ably access e en amily.
D.2 D.2 Memo y En opy and Sca e ing Time Scale
In sca e ing ep esen a ion, ime e olu ion is
S(ω)
. Modula Hamil onian
K=−ln ∆
de e mines ela i e en opy. Aligning
K
wi h sca e ing phase and
κ(ω)
yields equi alence
o memo y en opy and ime scale.
20